Collection 2022 - T3 - WS1 - Non-Linear and High Dimensional Inference

Organizer(s) Aamari, Eddie ; Aaron, Catherine ; Chazal, Frédéric ; Fischer, Aurélie ; Hoffmann, Marc ; Le Brigant, Alice ; Levrard, Clément ; Michel, Bertrand

Date(s) 03/10/2022 - 07/10/2022

linked URL https://indico.math.cnrs.fr/event/7545/

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Linear methods for non-linear inverse problems

By Botond Szabo

We consider recovering an unknown function $f$ from a noisy observation of the solution $u_f$ to a partial differential equation of the type $\mathcal{L} u_f=c(f,u_f)$ for a differential operator $\mathcal{L}$, and invertible function $c$, i.e. $f=e(\mathcal{L}u_f)$. Examples include amongst others the time-independent Schrödinger equation $\frac{1}{2}\Delta u_f = u_ff$ and the heat equation with absorption term $\frac{d u_f}{dt}-\frac{1}{2}\Delta u_f=f$. We transform this problem into the linear inverse problem of recovering $\mathcal{L}u_f$ under Dirichlet boundary condition, and show that Bayesian methods (with priors placed either on $u_f$ or $\mathcal{L}u_f$) for this problem may yield optimal recovery rates not only for $u_f$, but also for $f$. We also derive frequentist coverage guarantees for the corresponding Bayesian credible sets. Adaptive priors are shown to yield adaptive contraction rates for $f$, thus eliminating the need to know the smoothness of this function. The results are illustrated by several numerical simulations.

Information about the video

Date of recording 05/10/2022
Date of publication 26/04/2024
Institution IHP
Language English
Audience Researchers, Graduate Students
Director(s) Alexandre Duplessis, Marco Perez
Format MP4
Venue Amphitheater Hermite, IHP

Citation data

DOI 10.57987/IHP.2022.T3.WS1.008
Cite this video Szabo, Botond (05/10/2022). Linear methods for non-linear inverse problems. IHP. Audiovisual resource. DOI: 10.57987/IHP.2022.T3.WS1.008
URL https://dx.doi.org/10.57987/IHP.2022.T3.WS1.008